We show that in the worst case, Omega(nd) sidedness queries are required
to determine whether a set of n points in Rd is affinely
degenerate, i.e., whether it contains d+1 points on a common hyperplane.
This matches known upper bounds. We give a straightforward adversary argument, based on
the explicit construction of a point set containing Omega(nd)
``collapsible'' simplices, any one of which can be made degenerate without changing the
orientation of any other simplex. As an immediate corollary, we have an
Omega(nd) lower bound on the number of sidedness queries required to
determine the order type of a set of n points in Rd.
Using similar techniques, we also show that Omega(nd+1)
in­sphere queries are required to decide the existence of spherical degeneracies
in a set of n points in Rd.